Washer Method
The Washer Method is a technique used in calculus to find the volume of a solid of revolution
The Washer Method is a technique used in calculus to find the volume of a solid of revolution. This method is specifically used when the shape being revolved is between two curves.
To understand the Washer Method, let’s consider a simple example. Suppose we have a region in the xy-plane bounded by two functions: the upper function is f(x) and the lower function is g(x), and we are interested in finding the volume of the solid of revolution when this region is rotated around the x-axis.
To compute the volume using the Washer Method, we divide the region into infinitely thin vertical “slices” or “washers” from x = a to x = b. Each washer has a thickness of delta x. The radius of each washer is determined by the difference between the upper and lower functions, specifically, the difference between f(x) and g(x).
The key idea behind the Washer Method is that the volume of each washer can be approximated as a cylindrical shell with an inner radius equal to g(x) and an outer radius equal to f(x). The height of each washer is determined by delta x.
Therefore, the volume of each washer can be calculated using the formula for the volume of a cylindrical shell:
V = π * [(f(x))^2 – (g(x))^2] * delta x
To find the total volume, we sum up the volumes of all the washers by integrating this expression over the interval [a, b]:
V_total = ∫[a, b] π * [(f(x))^2 – (g(x))^2] * dx
This integral expression will give us the volume of the solid of revolution.
It’s important to note that the limits of integration, [a, b], correspond to the values of x where the two curves intersect. This is because the solid of revolution is formed only within the region defined by these intersection points.
Overall, the Washer Method provides a powerful tool for finding the volume of solids of revolution when the shape being rotated is between two curves.
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